Solving linear programming problem with uncertainty : probability interval and random set parameters

Probability intervals and random sets are special cases of interval-valued probability measures. In this paper, we analyze the relationship between a probability interval and a random set. We provide a construction of a random set for a given probability interval. This construction is easier to apply than an existence construction in [5].

A tool for appraisements by a group of experts, the experton, is explained. A semantic valuation on 11 positions (endecadery logic) is used. This tool is peculiarly adapted to expert systems. The experton concept arises from theories of fuzzy sets, intervals of confidence and random sets. It is a very convenient tool to aggregate experts' opinions in such a way that a decrease of entropy is realized at the end of the process and not at the beginning. The T-norms and T-conorms are not linear. An experton is an extension of the concept of random set (probabilistic set) where an interval of probabilities is obtained for each level alpha in

Merging the local and global approaches to probabilistic satisfiability

This paper focuses on the mechanical model uncertainty quantification when the system parameters are in the form of probability box; for example, when the probability distribution of the system variable is specified by interval parameters, the variable is in the form of probability box. We develop an improved outer discretization method (IODM) for constructing random sets which provides a convenient mechanism for representing probability box. The method can increase the fineness of the random set’s approximation to the bounds of the probability box with a limited focal element. It is shown that the method is helpful for the uncertainty quantification when the model is computationally expensive.

A random set approach to the optimization of uncertain structures

We propose a Bayesian higher-order probability logic reasoning approach with interval probability parameters to the problem of making inference from conditional knowledge, which combines weak conditional probability and conditional event algebra for approximate inferences. We define the bound-limited weak conditional interval probabilities, the corresponding probabilistic description, and the multiplication rules of weak conditional probabilities for joint probability distribution, and use higher-order conditional event to resolve a discrepancy between logic and probability. By extending normal measurable space with conditional event, we bring logic consistent with probability in denoting conditional knowledge, and then transform a higher-order conditional event to normal events and corresponding logical joint events via conditional event algebra. Based on multiplication rules, we compute the quantitative values of the events with interval parameters, and evaluate the value of higher-order conditional event and finish reasoning process. An illustrative example of application of our method shows how we make inferences from conditional knowledge.

In this paper, we propose a new model for linear fractional programming problem with absolute value functions. The major contribution of this paper is that transformation of linear fractional programming problem into separate linear programming problems with some theorems and then solution of these problems by popular algorithm. This work also used to simplex type algorithms to arrive at an optimal solution for a linear programming problem with absolute value. Moreover, we compare this method with an existing method. Numerical experiments are also given to illustrate the assertions.

An inventory-theory-based interval stochastic programming method and its application to Beijing’s electric-power system planning

In order to accommodating crossing hazard curves, which are non- proportional hazards, we consider in this paper a generalized time-dependent logistic hazard survival model, which has a time-dependent term. The model is a wholly parametric competitor for the Cox proportional hazard model. We compare different procedures to compute confiddence intervals for the model parameters in presence of random censoring. Our simulation study focus on the study of the coverage probabilities of these different confidence intervals and on the significance levels of some hypothesis tests. We discovered that parametric and non-parametric resampling methods can be successfully used for hypothesis testing and generating precise confidence intervals for the parameters even on small and moderate sized samples.

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

In this paper, economic order quantity (EOQ) models without shortages for single item and multi-items are presented. Here, the holding cost of the item is a continuous function of the order quantity. The costs involved in this model are imprecise in nature. The main contributions of this research are as follows: The proposed EOQ model is discussed in two cases by describing the model in an uncertain environment. In case-1, EOQ models with fuzzy parameters (like ordering cost, holding cost, and unit product cost) are considered. Here all the fuzzy parameters are represented by trapezoidal fuzzy numbers. The said EOQ model is carried out by using the signed-distance method. In case-2, EOQ models with interval parameters (like ordering cost, holding cost, unit product cost, and the total money investment for the quantities) are considered. This proposed model is solved by using interval linear programming problem (ILPP) technique based on the best and the worst optimum values of the objective function. Numerical examples are given to exemplify the proposed model and also the results of different models are compared.KeywordsEOQTrapezoidal fuzzy numberInterval numberSigned-distance methodInterval linear programming problem (ILPP)

In this paper a method of designing a structure with interval parameters and fuzzy set parameters is presented. This paper also outlines a procedure for designing a structure with random set parameters. All procedures use solutions of the interval equations which are based on the earlier works of both authors. Safety of the structures is determined by using interval limit state equations.

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.