A new linear ordering of fuzzy numbers on subsets of {{\mathcal F}({\pmb{\mathbb{R}}}})

We introduce a novel linear order on every family of fuzzy numbers which satisfies the assumption that their modal values must be all different and must form a compact subset of . A distinct new feature is that our linear determined procedure employs the corresponding order of a class interval associated with a confidence measure which seems intuitively anticipated. It is worthy noting that although we start from an entirely different rationale, we introduce a fuzzy ordering which initially coincides with the one established earlier by Ramik and Rimanek. However, this fuzzy ordering does not apply when the supports of the fuzzy numbers overlap. In order to cover such cases we extent this initial fuzzy ordering to the “extended fuzzy order” (XFO). This new XFO method includes a possibility and a necessity measure which are compared with the widely accepted PD and NSD indices of D. Dubois and H. Prade. The comparison shows that our possibility and necessity measures comply better with our intuition.     

Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility

This paper considers Stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random two-level linear programming problems. On the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal, together with the introduction of probability maximization criterion in stochastic programming, we propose new two-level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values. Through the proposed models, it is shown that the original two-level linear programming problems with fuzzy random variables can be transformed into deterministic two-level linear fractional programming problems. For the transformed problems, extended concepts of Stackelberg solutions are defined and computational methods are also presented. A numerical example is provided to illustrate the proposed methods.     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

Possibility linear programming with trapezoidal fuzzy numbers

Fuzzy linear programming with trapezoidal fuzzy numbers (TrFNs) is considered and a new method is developed to solve it. In this method, TrFNs are used to capture imprecise or uncertain information for the imprecise objective coefficients and/or the imprecise technological coefficients and/or available resources. The auxiliary multi-objective programming is constructed to solve the corresponding possibility linear programming with TrFNs. The auxiliary multi-objective programming involves four objectives: minimizing the left spread, maximizing the right spread, maximizing the left endpoint of the mode and maximizing the middle point of the mode. Three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. An investment example and a transportation problem are presented to demonstrate the implementation process of this method. The comparison analysis shows that the fuzzy linear programming with TrFNs developed in this paper generalizes the possibility linear programming with triangular fuzzy numbers.     

Insoluble multicriteria linear programming problems

This note concerns a method for analyzing insoluble multicriteria linear programming problems.     

A geometric approach for solving fuzzy linear programming problems

In this paper we first recall some definitions and results of fuzzy plane geometry, and then introduce some definitions in the geometry of two-dimensional fuzzy linear programming (FLP). After defining the optimal solution based on these definitions, we use the geometric approach for obtaining optimal solution(s) and show that the algebraic solutions obtained by Zimmermann method (ZM) and our geometric solutions are the same. Finally, numerical examples are solved by these two methods.     

Fuzzy inventory model with two warehouses under possibility measure on fuzzy goal

Multi-item inventory model with stock-dependent demand and two-storage facilities is developed in fuzzy environment (purchase cost, investment amount and storehouse capacity are imprecise) under inflation and time value of money. Joint replenishment and simultaneous transfer of items from one warehouse to another is proposed using basic period (BP) policy. As some parameters are fuzzy in nature, objective (average profit) function as well as some constraints are imprecise in nature. Model is formulated as to optimize the possibility/necessity measure of the fuzzy goal of the objective function and constraints are satisfied with some pre-defined necessity. A genetic algorithm (GA) is developed with roulette wheel selection, binary crossover and mutation and is used to solve the model when the equivalent crisp form of the model is available. In other cases fuzzy simulation process is proposed to measure possibility/necessity of the fuzzy goal as well as to check the constraints of the problem and finally the model is solved using fuzzy simulation based genetic algorithm (FSGA). The models are illustrated with some numerical examples and some sensitivity analyses have been done.     

Weighting method for bi-level linear fractional programming problems

In this paper we consider the solution of a bi-level linear fractional programming problem (BLLFPP) by weighting method. A non-dominated solution set is obtained by this method. In this article decision makers (DMs) provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control. We convert the hierarchical system into scalar optimization problem (SOP) by finding proper weights using the analytic hierarchy process (AHP) so that objective functions of both levels can be combined into one objective function. Here the relative weights represent the relative importance of the objective functions.     

A novel method for solving linear programming problems with symmetric trapezoidal fuzzy numbers

Linear programming (LP) is a widely used optimization method for solving real-life problems because of its efficiency. Although precise data are fundamentally indispensable in conventional LP problems, the observed values of the data in real-life problems are often imprecise. Fuzzy sets theory has been extensively used to represent imprecise data in LP by formalizing the inaccuracies inherent in human decision-making. The fuzzy LP (FLP) models in the literature generally either incorporate the imprecisions related to the coefficients of the objective function, the values of the right-hand-side, and/or the elements of the coefficient matrix. We propose a new method for solving FLP problems in which the coefficients of the objective function and the values of the right-hand-side are represented by symmetric trapezoidal fuzzy numbers while the elements of the coefficient matrix are represented by real numbers. We convert the FLP problem into an equivalent crisp LP problem and solve the crisp problem with the standard primal simplex method. We show that the method proposed in this study is simpler and computationally more efficient than two competing FLP methods commonly used in the literature.     

Fuzzy goal programming approach to multilevel programming problems

The purpose of this paper is to propose a procedure for solving multilevel programming problems in a large hierarchical decentralized organization through linear fuzzy goal programming approach. Here, the tolerance membership functions for the fuzzily described objectives of all levels as well as the control vectors of the higher level decision makers are defined by determining individual optimal solution of each of the level decision makers. Since the objectives are potentially conflicting in nature, a possible relaxation of the higher level decision is considered for avoiding decision deadlock. Then fuzzy goal programming approach is used for achieving highest degree of each of the membership goals by minimizing negative deviational variables. Sensitivity analysis with variation of tolerance values on decision vectors is performed to present how the solution is sensitive to the change of tolerance values. The efficiency of our concept is ascertained by comparing results with other fuzzy programming approaches.     

Sensitivity analysis in interval-valued trapezoidal fuzzy number linear programming problems

The aim of this article is to introduce a formulation of fuzzy linear programming problems involving the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy numbers as parameters. Indeed, such a formulation is the general form of trapezoidal fuzzy number linear programming problems. Then, it is demonstrated that study of the sensitivity analysis for the level (h^L,h^U)$(h^L\text{,}{h}^U)$-interval-valued trapezoidal fuzzy number linear programming problems gives rise to the same expected results as those obtained for trapezoidal fuzzy number linear programming problems.     

A linear programming-based optimization algorithm for solving nonlinear programming problems

In this paper a linear programming-based optimization algorithm called the Sequential Cutting Plane algorithm is presented. The main features of the algorithm are described, convergence to a Karush–Kuhn–Tucker stationary point is proved and numerical experience on some well-known test sets is showed. The algorithm is based on an earlier version for convex inequality constrained problems, but here the algorithm is extended to general continuously differentiable nonlinear programming problems containing both nonlinear inequality and equality constraints. A comparison with some existing solvers shows that the algorithm is competitive with these solvers. Thus, this new method based on solving linear programming subproblems is a good alternative method for solving nonlinear programming problems efficiently. The algorithm has been used as a subsolver in a mixed integer nonlinear programming algorithm where the linear problems provide lower bounds on the optimal solutions of the nonlinear programming subproblems in the branch and bound tree for convex, inequality constrained problems.     

Modeling centrality measures in social network analysis using bi-criteria network flow optimization problems

Centrality measures play an important role in the field of network analysis. In the particular case of social networks, the flow represents the way in which information passes through the network nodes. Freeman et al. (1991) were the first authors to relate centrality measures to network flow optimization problems in terms of betweenness, closeness, and the influence of one node over another one. Such measures are single dimensional and, in general, they amalgamate several heterogeneous dimensions into a single one, which is not suitable for dealing with most real-world problems. In this paper we extend the betweenness centrality measure (or concept) to take into account explicitly several dimensions (criteria). A new closeness centrality measure is defined to deal not only with the maximum flow between every ordered pair of nodes, but also with the cost associated with communications. We shall show how the classical measures can be enhanced when the problem is modeled as a bi-criteria network flow optimization problem.     

Fuzzy expectation values in multistage optimization problems

The study demonstrates the use of fuzzy expectation values in problems of multistage optimization under uncertainty. A practicable procedure is presented for the case where the optimization objective can be decomposed into a series of single-stage decision goals. Instead of probability theory, the uncertainty resolution is accomplished by fuzzy expectation values. In essence, then, the risk aversion is emboided in the selection of the fuzzy integration measure. If for example, the primary goal of the optimization is to achieve a strict cost minimum, then in the lack of information, a weaker goal can be introduced that corresponds to balancing the anticipated costs to the risk of exceeding these in reality. In a number of common optimization problems the method proposed facilitates a rapid solution with clear information on the risk involved.     

Test problem construction for linear bilevel programming problems

A method of constructing test problems for linear bilevel programming problems is presented. The method selects a vertex of the feasible region, far away from the solution of the relaxed linear programming problem, as the global solution of the bilevel problem. A predetermined number of constraints are systematically selected to be assigned to the lower problem. The proposed method requires only local vertex search and solutions to linear programs.     

Fuzzy approaches for multiple objective linear fractional optimization

The problem of finding a solution to a multiple objective linear fractional program arises in several real world situations.In this paper we advocate that fuzzy sets theory provides a basis for solving this problem with sufficient consistency and rigorousness.After representing imprecise aspirations of the decision maker by structured linguistic variables or converting the original problem via approximations or change of variables into a multiple objective linear program, techniques of fuzzy linear programming may be used to reach a satisfactory solution.It is shown that under reasonable restrictions, this solution is efficient (Pareto optimal) for the original problem. Numerical examples are also included for illustration.     

求解弱线性双层规划问题的一种全局优化方法

双层规划在经济、交通、生态、工程等领域有着广泛而重要的应用.目前对双层规划的研究主要是基于强双层规划和弱双层规划.然而,针对弱双层规划的求解方法却鲜有研究.研究求解弱线性双层规划问题的一种全局优化方法,首先给出弱线性双层规划问题与其松弛问题在最优解上的关系,然后利用线性规划的对偶理论和罚函数方法,讨论该松弛问题和它的罚问题之间的关系.进一步设计了一种求解弱线性双层规划问题的全局优化方法,该方法的优势在于它仅仅需要求解若干个线性规划问题就可以获得原问题的全局最优解.最后,用一个简单算例说明了所提出的方法是可行的.     

Fuzzy stochastic goal programming problems

In this paper, we present a model to measure attainment value of fuzzy stochastic goals. Then, the new measure is used to de-randomize and de-fuzzify the fuzzy stochastic goal programming problem and obtain a standard linear program (LP). A numerical example is provided to illustrate the proposed method.     

A linear programming approach to test efficiency in multi-objective linear fractional programming problems

In a multi-objective linear fractional programming problem (MOLFPP), it is often useful to check the efficiency of a given feasible solution, and if the solution is efficient, it is useful to check strong or weak efficiency. In this paper, by applying a geometrical interpretation, a linear programming approach is achieved to test weak efficiency. Also, in order to test strong efficiency for a given weakly efficient point, a linear programming approach is constructed.     

Some duality results on linear programming problems with symmetric fuzzy numbers

Recently, linear programming problems with symmetric fuzzy numbers (LPSFN) have considered by some authors and have proposed a new method for solving these problems without converting to the classical linear programming problem, where the cost coefficients are symmetric fuzzy numbers (see in [4]). Here we extend their results and first prove the optimality theorem and then define the dual problem of LPSFN problem. Furthermore, we give some duality results as a natural extensions of duality results for linear programming problems with crisp data.