Monge Property and Bounding Multivariate Probability Distribution Functions with Given Marginals and Covariances

The objective of this paper is to demonstrate the practical construction of a multivariate probability distribution function using an actual soil database containing su(CIUC), OCR, and four piezocone parameters. Five hundred and thirty-five multivariate data points were compiled from 40 clay sites around the world (Brazil, Canada, Hong Kong, Italy, Malaysia, Norway, Singapore, Sweden, UK, USA, and Venezuela). It was found that a multivariate probability distribution can be constructed by transforming each component of a multivariate normal distribution to a Johnson distribution. Existing bivariate regression equations focus on strong correlations. Weak correlations are typically discarded. Site investigation is a costly exercise and ideally, one should exploit all measured geotechnical data for design. The multivariate distribution is a concise model to summarize all available information. Conditional distributions can be easily derived to update the marginal distribution of any one parameter or the multivariate distribution of any group of parameters given information from other parameters. One of the objectives of site investigation is to perform cost-effective field tests and to evaluate design parameters based on these field measurements. Clearly, conditioning involving updating one or more design parameters using one or more field measurements is a natural probabilistic generalization of the current soil property evaluation methodology. INTRODUCTION When multivariate geotechnical data exist in sufficient amount, it is of significant practical usefulness to construct a multivariate probability distribution function. The applications include: (a) deriving the mean and coefficient of variation (COV) of any parameter given the information contained in a subset with possibly more than one parameter, and (b) evaluating if new strong pairwise correlations can be found either among the original components or some derived components. For the former, it is likely for the COV of a design parameter, say the undrained shear strength (su), to decrease when other parameters, say normalized cone tip resistance [(qtv)/ v] and overconsolidation (OCR), have been measured. This aspect is significant for reliability-based design. In fact, COV reduction can be viewed as a measure of the value of information and may eventually provide a sensible method for deciding if it is worthwhile to measure an additional parameter. For the latter, the ability to predict the existence of new correlations not included as part of model calibration provides a stronger scientific underpinning to correlation studies in geotechnical engineering. The reason is that these predictions can be falsified by taking new observations, which is the cornerstone of the scientific method. In other words, it is a lot harder to develop multivariate models, but if they do stand the test of time, they are usually more robust than bivariate models. The objective of this paper is to demonstrate the practical construction of a multivariate probability distribution function using an actual soil database containing su(CIUC), OCR, and

The traditional linear programming model is deterministic. The way that uncertainty is handled is to compute the range of optimality. After the optimal solution is obtained, typically by the simplex method, one considers the effect of varying each objective function coefficient, one at a time. This yields the range of optimality within which the decision variables remain constant. This sensitivity analysis is useful for helping the analyst get a sense for the problem. However, it is unrealistic because objective function coefficients tend not to stand still. They are typically profit contributions from products sold and are subject to randomly varying selling prices. In this paper, a realistic linear program is created for simultaneously randomizing the coefficients from any probability distribution. Furthermore, we present a novel approach for designing a copula of random objective function coefficients according to a specified rank correlation. The corresponding distribution of objective function values is created. This distribution is examined directly for central tendency, spread, skewness and extreme values for the purpose of risk analysis. This enables risk analysis and business analytics, emerging topics in education and preparation for the knowledge economy.

Many problems appear to be arising in the present as a result of variations in transportation networks. The stochastic fixed-charge transportation problem (SFCTP) is one such problem. The SFCTP is transformed into a chance-constrained programming (CCP) problem where supply and demand are stochastic and objective functions are in a rough interval. In this model, to analyze the multi-objective rough interval stochastic fixed-charge transportation problem (MORISFCTP), where the objective function coefficients are represented by rough intervals and the supply and destination factors are probabilistic constraints. This model operates an expected value operator to deal with uncertainty, in which the coefficient of the objective functions in the fuzzy is changed to a crisp form, and the probabilistic constraints are converted to a deterministic form by the Weibull distribution. To produce the optimal compromise solutions of the proposed model, three distinct methods are used: the fuzzy programming approach, the method of a linear weighted sum, and the €-constraint method. Lastly, the paper delivers a practical illustration of a MORISFCTP to demonstrate the usefulness and feasibility of the suggested methodology.

In recent years, many researchers in the field of fuzzy mathematics and operations research in the world are interested to solve the interval type-2 fuzzy linear programming (IT2FLP) problems. In IT2FLP problems all parameters, including coefficients in objective function, coefficients in the constraints and right hand sides of the constraints are interval type-2 fuzzy numbers (IT2FN). Hence, the IT2FLP model is one of the most important and the most appropriate models to interpret and determine the optimal decision. In this paper, a new ranking function method is proposed for solving these problems. Moreover, an example is used to study and describe the proposed method.

Fuzzy preference relations in models of decision making

Linear program under changes in the system matrix coefficients has proved to be more complex than changes of the coefficients in objective functions and right hand sides. The most of the previous studies deals with problems where only one coefficient, a row (column), or few rows (columns) are linear functions of a parameter. This work considers a more general case, where all the coefficients are polynomial (in the particular case linear) functions of the parameter tT??R. For such problems, assuming that some non-singularity conditions hold and an optimal base matrix is known for some particular value t of the parameter, corresponding explicit optimal basic solution in the neighborhood of t is determined by solving an augmented LP problem with real system matrix coefficients. Parametric LP can be utilized for example to model the production problem where, technology, resources, costs and similar categories vary with time. .

In this paper, a linear fractional programming problem with interval coefficients in both objective function and constraints is solved. The uncertain objective function is transformed into two deterministic objective functions. The two objective functions are converted into a single objective problem through the linear combination method. Through a modified possibility degree, the uncertain inequality and equality constraints are transformed into deterministic inequality constraints. The existing results are concerning the qualitative and quantitative analysis of basic notions in parametric linear fractional programming problem. These notions are the set of feasible parameters, the solvability set and the stability set of the first kind. A numerical example is investigated to demonstrate the effectiveness of the present approach.

A nonlinear interval number programming method for uncertain optimization problems

<p>Any model we use is an approximation of the real world and associated with some uncertainty about the transition of the model result into the real world. This issue is even stressed in situations where models are used to predict a set of variables over time. A prominent and widely used example are runs of climate models that generate ensembles of possible future paths of our climate including several correlated variables. Typically, these ensembles are presented and assessed in an univariate approach where e.g. spaghetti plots of single variables depict the variability within the future paths. What remains hidden are the dependencies among the variables. As the correlation measures might be ambiguous and summarize the possibly complex dependence structures in a single value, we use the concept of copulas to illustrate the variability of multivariate distributions. Going back to Sklar’s theorem any continuous multivariate distribution can be decomposed into its univariate margins and their copula describing the dependence between the univariate marginals. Copulas can serve two purposes in this context, to model and quantify i) the multivariate variability within the ensemble and ii) the variability in the dependence between the variables among the ensembles.</p><p>We illustrate the effect of different copula families on the inherent uncertainties in the ensemble based on synthetic data. Furthermore, we use climate predictions of the current century to identify and study the dependence and its variability in the data set. In order to increase the data and to avoid the expectation that we can do an assessment precisely for each year, the data is grouped by decades. Our approach is illustrated in R and an exploratory analysis is supported through an interactive Shiny application. The Shiny application also serves to communicate the multivariate uncertainty in the data.</p>

In this paper we present an overview about the recently developed theory of discrete moment problems, i.e., moment problems where the supports of the random variables involved are discrete. We look for the minimum or maximum of a linear functional acting on an unknown probability distribution subject to a finite number of moment constraints. Using linear programming methodology, we present structural theorems, in both the univariate and multivariate cases, for the dual feasible bases and show how the relevant problems can be solved by suitable adaptations of the dual method. The condition on the objective function is a kind of higher order convexity, expressed in terms of divided differences. A variant of the above, the discrete binomial moment problem, as well as generalization for discrete variable Chebyshev systems are also discussed. Finally, we present novel applications to valuations of financial instruments.

A distribution P 0 of a random vector X = ( X 1 ,..., X n ) is said to be linear exponential in a parameter θ = ( θ 1 ,..., θ n ) if [Formula: see text]. A multivariate linear exponential distribution is binomial if and only if its univariate marginals are binomial.

This paper studies the problem of tangible assets acquisition within the company by proposing a new hybrid model that uses linear programming and fuzzy numbers. Regarding linear programming, two methods were implemented in the model, namely: the graphical method and the primal simplex algorithm. This hybrid model is proposed for solving investment decision problems, based on decision variables, objective function coefficients, and a matrix of constraints, all of them presented in the form of triangular fuzzy numbers. Solving the primal simplex algorithm using fuzzy numbers and coefficients, allowed the results of the linear programming problem to also be in the form of fuzzy variables. The fuzzy variables compared to the crisp variables allow the determination of optimal intervals for which the objective function has values depending on the fuzzy variables. The major advantage of this model is that the results are presented as value ranges that intervene in the decision-making process. Thus, the company’s decision makers can select any of the result values as they satisfy two basic requirements namely: minimizing/maximizing the objective function and satisfying the basic requirements regarding the constraints resulting from the company’s activity. The paper is accompanied by a practical example.

This paper deals with stability analysis in multi-objective combinatorial optimization problems. The stability radius of an efficient solution is defined as the maximal adjustment of the problem parameters such that this solution remains efficient. An algorithm based on inverse optimization is proposed to compute it. The adjustment is limited to the coefficients of the objective functions and measured by the Chebyshev norm. This approach is applied to randomly generated instances of the bi-objective knapsack problem and computational results are reported. Several illustrative examples are analyzed.

This paper considers interactive fuzzyprogramming for multi-level 0-1 programming problems involving random variable coefficients both in objective functions and constraints.Following the concept of fractile criterion optimization together with chance constrained programming, the formulated stochastic multi-level 0-1 programming problems are transformed into deterministic ones.Taking into account vagueness of judgments of the decision makers, interactive fuzzy programming is presented.In the proposed interactive method, after determining the fuzzy goals of the decision makers at all levels, a satisfactory solution is derived efficiently by updating satisfactory levels of the decision makers with considerations of overall satisfactory balance among all levels.For solving the transformed deterministic problems efficiently, tabu search for general 0-1 programming problems is introduced.An illustrative numerical example for a three-level 0-1 programming problem is provided to clarify the proposed method.

Optimistic Stackelberg solutions to bilevel linear programming with fuzzy random variable coefficients