Approximation of Probabilistic Constraints in Stochastic Programming Problems with a Probability Measure Kernel

The work is devoted to the development of a method for solving the stochastic programming problem with a deterministic objective function and individual probabilistic constraints. Each probabilistic constraint is a constraint on the probability of inequality for a certain loss function that is linear on random parameters. In this case, the loss function may be non-linear in strategies. It is proposed to replace each probabilistic constraint by an equivalent inequality for the quantile function. This inequality is approximated using the notion of the probability measure kernel. The kernel is defined as the intersection of all closed confidence half-spaces. It is known that if the kernel satisfies the regularity property and the loss function is linear in random parameters then the quantile function can be found as the maximum of the loss function in realizations of random parameters on the probability measure kernel. To evaluate quantiles an external polyhedral approximation [1] of the probability measure kernel is used. When replacing a kernel by its approximation the maximum mentioned above is an upper estimate of the exact value of the quantile function. As a result, each quantile constraint is replaced by several deterministic inequalities.

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

In this paper, we consider a Multi-Objective Stochastic Interval-Valued Linear Fractional Integer Programming problem (MOSIVLFIP). We especially deal with a multi-objective stochastic fractional problem involving an inequality type of constraints, where all quantities on the right side are log-normal random variables, and the objective functions coefficients are fractional intervals. The proposed solving procedure is divided in three steps. In the first one, the probabilistic constraints are converted into deterministic ones by using the chance constrained programming technique. Then, the second step consists of transforming the studied problem objectives on an optimization problem with an interval-valued objective functions. Finally, by introducing the concept of weighted sum method, the equivalent converted problem obtained from the two first steps is transformed into a single objective deterministic fractional problem. The effectiveness of the proposed procedure is illustrated through a numerical example.

Consideration was given to the two-stage problem of stochastic linear programming with a discrete distribution of the random parameter vector. The property of continuity of the quantile function in strategy was proved, the sufficient conditions for existence of solution were formulated, and an algorithm to determine the guaranteeing solution was constructed on the basis of the confidence method and the duality theorem. A deterministic equivalent of the considered problem in the form of a linear programming problem was given for the scalar case.

A possibilistic linear program is equivalent to a stochastic linear program in a special case

The concept of ranking method is an efficient approach to rank fuzzy numbers. In this paper, we have studied stochastic fuzzy multiobjective linear fractional programming problem (SFMOLFPP) where SFMOLFPP is transformed to its equivalent deterministic-crisp multiobjective linear programming problem (MOLPP). To study SFMOLFPP, a SFMOLFPP is presented in which the fuzzy coefficients and scalars in the linear fractional objectives and the fuzzy coefficients are characterised by triangular or trapezoidal fuzzy numbers. The left hand side of the stochastic fuzzy constraints are characterised by triangular or trapezoidal fuzzy numbers, while the right hand sides are assumed to be independent random variable with known distribution function. We have modify Iskander's approach [16] to transform the suggested problem to its equivalence deterministic-crisp MOLPP. We have also used ranking function in SFMOLFPP to find the pareto optimal solution of the reduced multiobjective linear fractional programming problem (MOLFPP). One numerical example is presented to demonstrate two methodologies.

An effective heuristic for multistage linear programming with a stochastic right-hand side

The study of making the best decision under risk management in a variety of areas of our lives is known as Stochastic Programming. We will go through two-stage Stochastic Linear Programming approaches for a variety of real-world choice issues, as well as how to solve them. We will achieve this by constructing stochastic linear programming models based on real-world situations like the well-known Farmer's situation and News Vendors problems. The influence of pricing, Stochastic Integer Linear Programming problem, second stage Stochastic Integer Linear Programming problem, first stage Stochastic Binary Linear Programming problem, risk aversion problem, and continuous function for random variables based on two-stage SLP with the aid of Farmer's problem will all be examined. We will address the Newsvendor’s problem with Deterministic Equivalent Stochastic Linear Programming, an extension of Deterministic Stochastic Linear Programming for risk aversion with a high number of decision variables and restrictions, utilizing the two-stage Stochastic Linear Programming approach once more. Hand calculation is a challenging way to acquire the solution to the problems. As a result, we will use the programming language AMPL to design computer solutions for tackling both farmer and newsvendor difficulties. We will also utilize MATLAB to create graphs for the farmer's problem's continuous function. Dhaka Univ. J. Sci. 72(1): 30-45, 2024 (January)

In Operations Research, linear-fractional programming is considered as the generalization of linear programming problem. While in a linear programming the objective function is a linear function, and in a linear-fractional programming the objective function is the ratio of two linear or non-linear functions. The majority of the algorithm used for solving the linear fractional programming problem relies upon the classical simplex method. In this paper, we have proposed a new algorithm for solving a linear fractional programming problem in which the objective function is a combination of linear fractional function, while constraint functions are in the form of linear inequalities. Our proposed algorithm is based on the extension of the method, which is used to solve linear programming problems with linear constraints. The primary intent behind developing this method is that we did not need to transform the linear fractional programming problem into linear programming problem, and also it helps in finding out the feasible region via a sequence of points in the direction that improves the feasibility of the fractional objective function. Numerical examples are given to illustrate the use of these proposed methods. Lastly, to demonstrate the efficacy of the proposed algorithm, we have compared the findings obtained with other approaches to display our algorithm's efficacy.

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ABSTRACTThe computational complexity of linear and nonlinear programming problems depends on the number of objective functions and constraints involved and solving a large problem often becomes a difficult task. Redundancy detection and elimination provides a suitable tool for reducing this complexity and simplifying a linear or nonlinear programming problem while maintaining the essential properties of the original system. Although a large number of redundancy detection methods have been proposed to simplify linear and nonlinear stochastic programming problems, very little research has been developed for fuzzy stochastic (FS) fractional programming problems. We propose an algorithm that allows to simultaneously detect both redundant objective function(s) and redundant constraint(s) in FS multi-objective linear fractional programming problems. More precisely, our algorithm reduces the number of linear fuzzy fractional objective functions by transforming them in probabilistic–possibilistic constraints characterized by predetermined confidence levels. We present two numerical examples to demonstrate the applicability of the proposed algorithm and exhibit its efficacy.

A study on multiobjective stochastic linear programming (MSLP) problems with partial information on probability distribution is conducted. A method is proposed to utilise the concept of dominated solution for the multiobjective linear programming (MLP) problems, and find a pareto optimal solution (POS) without converting the MLP problem into its unique linear programming (LP) problem. An algorithm is proposed along with a numerical example which illustrated the practicability of the proposed algorithm. Comparison of results with existing methods shows the efficiency of the proposed method based on the analysis of results performed.

A technique is developed for finding a closed form expression for the cumulative distribution function of the maximum value of the objective function in a stochastic linear programming problem, where either the objective function coefficients or the right hand side coefficients are continuous random vectors with known probability distributions. This is the “wait and see” problem of stochastic linear programming. Explicit results for the distribution problem are extremely difficult to obtain; indeed, previous results are known only if the right hand side coefficients have an exponential distribution [1]. To date, no explicit results have been obtained for stochastic c, and no new results of any form have appeared since the 1970’s. In this paper, we obtain the first results for stochastic c, and new explicit results if b an c are stochastic vectors with an exponential, gamma, uniform, or triangle distribution. A transformation is utilized that greatly reduces computational time.

We present a parallelization of the revised simplex method for large extensive forms of two-stage stochastic linear programming (LP) problems. These problems have been considered too large to solve with the simplex method; instead, decomposition approaches based on Benders decomposition or, more recently, interior-point methods are generally used. However, these approaches do not provide optimal basic solutions, which allow for efficient hot-starts (e.g., in a branch-and-bound context) and can provide important sensitivity information. Our approach exploits the dual block-angular structure of these problems inside the linear algebra of the revised simplex method in a manner suitable for high-performance distributed-memory clusters or supercomputers. While this paper focuses on stochastic LPs, the work is applicable to all problems with a dual block-angular structure. Our implementation is competitive in serial with highly efficient sparsity-exploiting simplex codes and achieves significant relative speed-ups when run in parallel. Additionally, very large problems with hundreds of millions of variables have been successfully solved to optimality. This is the largest-scale parallel sparsity-exploiting revised simplex implementation that has been developed to date and the first truly distributed solver. It is built on novel analysis of the linear algebra for dual block-angular LP problems when solved by using the revised simplex method and a novel parallel scheme for applying product-form updates.

OF THE DISSERTATION Properties and solutions of a class of stochastic programming problems with probabilistic constraints by Kunikazu Yoda Dissertation Director: Andras Prekopa We consider two types of probabilistic constrained stochastic linear programming problems and one probability bounding problem. The first type involves a random left-hand side matrix whose rows are independent and normally distributed. The quasi-concavity of the constraining function needed for the convexity of the problem is ensured if the factors of the function are uniformly quasi-concave. A necessary and sufficient condition is given for that property to hold. We show practical application in optimal portfolio construction. The second type is the stochastic multidimensional knapsack problem which involves a random left-hand side matrix with independent components and 0-1 decision variables. We show that the problem is convex, under some condition on the parameters, for special continuous and discrete distributions: gamma, normal, Poisson, and binomial. Numerical experiments suggest that the problem can be solved as efficiently as its deterministic version for moderate sized instances. In the last problem, we formulate the linear programming problems that give improved lower and upper bounds on the probability of the union of events when the probabilities of some individual or intersections of events in a first few terms of the inclusion-exclusion principle are 0 or very small.