A new Chance-Variance optimization criterion for portfolio selection in uncertain decision systems

The Markowitz’s mean-variance (M-V) model has received widespread acceptance as a practical tool for portfolio optimization, and his seminal work has been widely extended in the literature. The aim of this article is to extend the M-V method in hybrid decision systems. We suggest a new Chance-Variance (C-V) criterion to model the returns characterized by fuzzy random variables. For this purpose, we develop two types of C-V models for portfolio selection problems in hybrid uncertain decision systems. Type I C-V model is to minimize the variance of total expected return rate subject to chance constraint; while type II C-V model is to maximize the chance of achieving a prescribed return level subject to variance constraint. Hence the two types of C-V models reflect investors’ different attitudes toward risk. The issues about the computation of variance and chance distribution are considered. For general fuzzy random returns, we suggest an approximation method of computing variance and chance distribution so that C-V models can be turned into their approximating models. When the returns are characterized by trapezoidal fuzzy random variables, we employ the variance and chance distribution formulas to turn C-V models into their equivalent stochastic programming problems. Since the equivalent stochastic programming problems include a number of probability distribution functions in their objective and constraint functions, conventional solution methods cannot be used to solve them directly. In this paper, we design a heuristic algorithm to solve them. The developed algorithm combines Monte Carlo (MC) method and particle swarm optimization (PSO) algorithm, in which MC method is used to compute probability distribution functions, and PSO algorithm is used to solve stochastic programming problems. Finally, we present one portfolio selection problem to demonstrate the developed modeling ideas and the effectiveness of the designed algorithm. We also compare the proposed C-V method with M-V one for our portfolio selection problem via numerical experiments.     

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.     

模糊机会约束数据包络分析模型及其应用

为求解模糊环境下输出倾向的数据包络分析（DEA）模型,利用可信性测度建立了一类新的输出倾向的可信性DEA（CDEA）模型,其中目标函数采用了模糊机会约束规划的概念,且所有的约束条件中都含有模糊输入和模糊输出数据。当模糊输入和模糊输出数据为相互独立的梯形模糊变量时,把所建立的CDEA模型转化为其清晰等价形式,进而研究了模型的两个基本性质;通过一个应用实例来说明所建立CDEA模型的有效性。     

Stochastic simulation-based genetic algorithm for chance constraint programming problems with continuous random variables

In this article, we present a stochastic simulation-based genetic algorithm for solving chance constraint programming problems, where the random variables involved in the parameters follow any continuous distribution. Generally, deriving the deterministic equivalent of a chance constraint is very difficult due to complicated multivariate integration and is only possible if the random variables involved in the chance constraint follow some specific distribution such as normal, uniform, exponential and lognormal distribution. In the proposed method, the stochastic model is directly used. The feasibility of the chance constraints are checked using stochastic simulation, and the genetic algorithm is used to obtain the optimal solution. A numerical example is presented to prove the efficiency of the proposed method.     

Modeling fuzzy data envelopment analysis by parametric programming method

Data envelopment analysis (DEA) is a methodology for measuring the relative efficiency of decision making units (DMUs) consuming the same types of inputs and producing the same types of outputs. This paper studies the DEA models with type-2 data variations. In order to deal with the existed type-2 fuzziness, we propose the mean reduction methods for type-2 fuzzy variables. Based on the mean reductions of the type-2 fuzzy inputs and outputs, we formulate a new class of fuzzy generalized expectation DEA models. When the inputs and outputs are mutually independent type-2 triangular fuzzy variables, we discuss the equivalent parametric forms for the constraints and the generalized expectation objective, where the parameters characterize the degree of uncertainty of the type-2 fuzzy coefficients so that the information cannot be lost via our reduction method. For any given parameters, the proposed model becomes nonlinear programming, which can be solved by standard optimization solvers. To illustrate the modeling idea and the efficiency of the proposed DEA model, we provide one numerical example.     

Genetic algorithm-based fuzzy goal programming for class of chance-constrained programming problems

This paper presents a procedure for solving a multiobjective chance-constrained programming problem. Random variables appearing on both sides of the chance constraint are considered as discrete random variables with a known probability distribution. The literature does not contain any deterministic equivalent for solving this type of problem. Therefore, classical multiobjective programming techniques are not directly applicable. In this paper, we use a stochastic simulation technique to handle randomness in chance constraints. A fuzzy goal programming formulation is developed by using a stochastic simulation-based genetic algorithm. The most satisfactory solution is obtained from the highest membership value of each of the membership goals. Two numerical examples demonstrate the feasibility of the proposed approach.     

Minimum risk criterion for uncertain production planning problems

This paper considers a new class of multi-product source and multi-period fuzzy random production planning problems with minimum risk and service levels where both the demands and the production costs are assumed to be uncertain and characterized as fuzzy random variables with known distributions. The proposed problems are formulated as a fuzzy random production planning (FRPP) model by maximizing the mean chance of the total costs less than a given allowable investment level. Because the exact value of the objective function for a given decision variable cannot be easily obtained, we adopt an approximation approach (AA) to evaluate the objective value and then discuss the convergence of the AA, including the convergence of the objective value, the convergence of the optimal solutions and the convergence of the optimal value. Since the approximating multi-product source multi-period FRPP model is neither linear nor convex, an approximation-based hybrid monkey algorithm (MA) which combines the AA, stochastic simulation (SS), neural network (NN) and MA is designed to solve the proposed model. Finally, numerical examples are provided to illustrate the effectiveness of the hybrid monkey algorithm.     

An interactive fuzzy satisficing method based on variance minimization under expectation constraints for multiobjective stochastic linear programming problems

In this paper, we focus on multiobjective linear programming problems involving random variable coefficients in objective functions and constraints. Using the concept of chance constrained conditions, such multiobjective stochastic linear programming problems are transformed into deterministic ones based on the variance minimization model under expectation constraints. After introducing fuzzy goals to reflect the ambiguity of the decision maker??s judgements for objective functions, we propose an interactive fuzzy satisficing method to derive a satisficing solution for them as a fusion of the stochastic programming and the fuzzy one. The application of the proposed method to an illustrative numerical example shows its usefulness.     

INTERACTIVE FUZZY PROGRAMMING BASED ON FRACTILE CRITERION OPTIMIZATION MODEL FOR TWO-LEVEL STOCHASTIC LINEAR PROGRAMMING PROBLEMS

In this article, we focus on two-level linear programming problems involving random variable coefficients in objective functions and constraints. Following the concept of chance constrained programming, the two-level stochastic linear programming problems are transformed into deterministic ones based on the fractile criterion optimization model. After introducing fuzzy goals for objective functions, interactive fuzzy programming to derive a satisfactory solution for decision makers is presented as a fusion of a stochastic approach and a fuzzy one. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

Stochastic facility and transfer point covering problem with a soft capacity constraint

Existing models for transfer point location problems (TPLPs) do not guarantee the desired service time to customers. In this paper, a facility and TPLP is formulated based on a given service time that is targeted by a decision maker. Similar to real‐world situations, transportation times and costs are assumed to be random. In general, facilities are capacitated. However, in emergency services, they are not allowed to reject the customers for out of capacity reasons. Therefore, a soft capacity constraint for the facilities and a second objective to minimize the overtime in the facility with highest assigned demand are proposed. To solve the biobjective model with random variables, a variance minimization technique and chance‐constraint programming are applied. Thereafter, using fuzzy multiple objective linear programming, the proposed biobjective model is converted to a single objective. Computational results on 30 randomly designed experimental problems confirm satisfactory performance of the proposed model in reducing the variance of solutions as well as the overtime in the busiest facility.     

A goal programming approach to fuzzy linear regression with fuzzy input�Coutput data

Fuzzy linear regression is an active area of research. In the literature, fuzziness is considered in outputs and/or in inputs. This paper focuses on both fuzzy inputs and fuzzy outputs. First, some approximations for multiplication of two triangular fuzzy numbers are introduced. Then, to evaluate the fuzzy linear regression, the best approximation is selected to minimize a suitable function via goal programming. An important feature of the proposed model is that it takes into account the centers of fuzzy data as well as their spreads. Moreover, it is flexible to deal with both symmetric and non-symmetric data. Furthermore, it can handle the crisp inputs and trapezoidal fuzzy outputs easily. To show the efficiency of the proposed model, some numerical examples are solved and compared with some earlier methods.     

The upper and lower bound evaluation based on the quantile efficiency in stochastic data envelopment analysis

Data envelopment analysis (DEA) has been extended to handle random inputs and outputs by using chance constrained programming. In this paper, for DMUs with random inputs and outputs, we aim to measure a kind of relative efficiency, and achieve it from the optimistic viewpoint and the pessimistic viewpoint respectively. Considering the quantile of the distribution of the weighted output-input ratio of each DMU, we develop two stochastic DEA models to obtain the upper and lower bounds of the quantile efficiency under a constraint, and then achieve an interval efficiency evaluation. The best quantile efficiency and the worst quantile efficiency achieved by our models are closely similar to the CCR efficiency and belong to relative efficiencies. Further, the deterministic equivalents of our models are developed when the input and output vector of each DMU follows a multivariate joint normal distribution. Finally, three examples are presented to illustrate the performance of our approach.     

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.     

机会约束下贷款组合优化决策的方差最小化模型

通过把贷款的收益率刻画为模糊变量,提出了机会约束下贷款组合优化决策的方差最小化模型。针对贷款收益率是特殊的三角模糊变量的情况,给出模型的清晰等价类,对等价类模型用传统的方法进行求解。对于贷款收益率的隶属函数比较复杂的情况,应用集成模糊模拟、神经网络、遗传算法和同步扰动随机逼近算法的混合优化算法求解模型。数值算例验证了模型和算法的有效性。     

Fuzzy multicriterion design using immune network simulation

Structural optimization problems have been traditionally formulated in terms of crisply defined objective and constraint functions. With a shift in application focus towards more practical problems, there is a need to incorporate fuzzy or noncrisp information into an optimization problem statement. Such practical design problems often deal with the allocation of resources to satisfy multiple, and frequently conflicting design objectives. The present paper deals with a genetic algorithm based optimization procedure for solving multicriterion design problems where the objective or constraint functions may not be crisply defined. The approach uses a genetic algorithm based simulation of the biological immune system to solve the multicriterion design problem; fuzzy set theory is adopted to incorporate imprecisely defined information into the problem statement. A notable strength of the proposed approach is its ability to generate a Pareto-Edgeworth front of compromise solutions in a single execution of the GA. Received May 8, 2000     

Type-2 fuzzy multi-objective DEA model: An application to sustainable supplier evaluation

In the last 10 years, sustainable supply chain management (SSCM) has become one of the important topics in business and academe. Sustainable supplier performance evaluation and selection play a significant role in establishing an effective SSCM. One of the techniques that can be used for evaluating sustainable supplier performance is data envelopment analysis (DEA). The conventional DEA methods require accurate measurement of both input and output variables present in the problem. In practice, the observed values of the input and output data present in real-world problems are often imprecise. To cope with this situation, fuzzy DEA models were constructed for expressing relative fuzzy efficiencies of decision-making units (DMUs). However, fuzzy DEA is still limited to fuzzy input/output data while some inputs and outputs might be affected by various factors of uncertainty and information granularity, meaning that they could be better modeled in terms of fuzzy sets of type-2. In this paper, we develop a multi-objective DEA model in a setting of type-2 fuzzy modeling to evaluate and select the most appropriate sustainable suppliers. In the proposed model, both efficiency and effectiveness are considered to describe the integrated productivity of suppliers. In sequel, chance constrained programming, critical value-based reduction methods and equivalent transformations are considered to solve the problem. A detailed case study is employed to show the advantages of the proposed model in terms of measuring effectiveness, efficiency and productivity in an uncertain environment expressed at different confidence levels. At the same time, the results demonstrate that the model is capable of helping decision makers to balance economic, social, and environmental factors when selecting sustainable suppliers.     

Analytical structure and stability analysis of a fuzzy PID controller

Analytical structure for a fuzzy PID controller is introduced by employing two fuzzy sets for each of the three input variables and four fuzzy sets for the output variable. This structure is derived via left and right trapezoidal membership functions for inputs, trapezoidal membership functions for output, algebraic product triangular norm, bounded sum triangular co-norm, Mamdani minimum inference method, and center of sums (COS) defuzzification method. Conditions for bounded-input bounded-output (BIBO) stability are derived using the Small Gain Theorem. Finally, two numerical examples along with their simulation results are included to demonstrate the effectiveness of the simplest fuzzy PID controller.     

Computation of probabilistic linear programming problems involving normal and log-normal random variables with a joint constraint

A method for solving single- and multi-objective probabilistic linear programming problems with a joint constraint is presented. It is assumed that the parameters in the probabilistic linear programming problems are random variables, and the probabilistic problem is converted to an equivalent deterministic mathematical programming problem. In this paper the parameters are generally considered as normal and log-normal random variables. A non-linear programming method is used to solve the single-objective deterministic problem, and a fuzzy programming method is used to solve the multi-objective deterministic problem. Finally, a numerical example is presented to illustrate the methodology.     

A gamma type-2 defuzzification method for solving a solid transportation problem considering carbon emission

This paper intends to develop a multi-objective solid transportation problem considering carbon emission, where the parameters are of gamma type-2 fuzzy in nature. This paper proposed the defuzzification process for gamma type-2 fuzzy variable using critical value (CV ) and nearest interval approximation method. A chance constraint programming problem is generated using the CV based reduction method to convert the fuzzy problem to its equivalent crisp form. Applying the \(\alpha \)-cut based interval approximation method, a deterministic problem is developed. Some real life data are used to minimize the cost and carbon emission. LINGO standard optimization solver has been used to solve the multi-objective problem using weighted sum method and intuitionistic fuzzy programming technique. The Genetic Algorithm (GA) and Particle Swarm Optimization (PSO) algorithm are implemented to generate efficient optimal solution by converting the multi-objective problem to a single objective problem using penalty cost for carbon emission. After solving the problem, analysis on some particular cases has been presented. The sensitivity analysis has been shown to different credibility levels of cost, emission, source, demand, conveyance to find total cost, emission and transported amount in each level. A comparison study on the performance of three algorithms (LINGO, GA and PSO) is presented. At the end, some graphs have been plotted which shows the effect of emission with different emission parameters.     

On minimum-risk problems in fuzzy random decision systems

In a decision-making process, we may face a hybrid environment where linguistic and frequent imprecision nature coexists. The problem of frequent imprecision can be solved by probability theory, while the problem of linguistic imprecision can be tackled by possibility theory. Therefore, to solve this hybrid decision-making problem, it is necessary to combine both theories effectively. In this paper, we restrict our attention to this hybrid decision-making problem, where the input data are imprecise and described by fuzzy random variables. Fuzzy random variable is a mapping from a probability space to a collection of fuzzy variables, it is an appropriate tool to deal with twofold uncertainty with fuzziness and randomness in an optimization framework. The purpose of this paper is to present reasonable chances of a fuzzy random event characterized by fuzzy random variables so that they can connect with the expected value operators of a fuzzy random variable via Choquet integrals, just like the relation between the probability of a random event and the mathematical expectation of a random variable, and that between the credibility of a fuzzy event and the expected value operator of a fuzzy variable. Toward that end, we take fuzzy measure and fuzzy integral theory as our research tool, and present three kinds of mean chances of a fuzzy random event via Choquet integrals. After discussing the duality of the mean chances, we use the mean chances to define the expected value operators of a fuzzy random variable via Choquet integrals. To show the reasonableness of the mean chance approach, we prove the expected value operators defined in this paper coincide with those presented in our previous work. Using the mean chances, we present a new class of fuzzy random minimum-risk problems, where the objective and the constraints are all defined by the mean chances. To solve general fuzzy random minimum-risk optimization problems, a hybrid intelligent algorithm, which integrates fuzzy random simulations, genetic algorithm and neural network, is designed, and its feasibility and effectiveness are illustrated by numerical examples.