Randomized Objective Function Linear Programming in Risk Management

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. .

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.

For many of us, modern-day linear programming (LP) started with the work of George Dantzig in 1947. However, it must be said that many other scientists have also made seminal contributions to the subject, and some would argue that the origins of LP predate Dantzig’s contribution. It is matter open to debate [36]. However, what is not open to debate is Dantzig’s key contribution to LP computation. In contrast to the economists of his time, Dantzig viewed LP not just as a qualitative tool in the analysis of economic phenomena, but as a method that could be used to compute actual answers to specific real-world problems. Consistent with that view, he proposed an algorithm for solving LPs, the simplex algorithm [12]. To this day the simplex algorithm remains a primary computational tool in linear and mixed-integer programming (MIP). In [11] it is reported that the first application of Dantzig’s simplex algorithm to the solution of a non-trivial LP was Laderman’s solution of a 21 constraint, 77 variable instance of the classical Stigler Diet Problem [41]. It is reported that the total computation time was 120 man-days! The first computer implementation of an at-least modestly general version of the simplex algorithm is reported to have been on the SEAC computer at the then National Bureau of Standards [25]. (There were apparently some slightly earlier implementations for dealing with models that were “triangular”, that is, where all the linear systems could be solved by simple addition and subtraction.) Orchard-Hays [35] reports that several small instances having as many as 10 constraints and 20 variables were solved with this implementation. The first systematic development of computer codes for the simplex algorithm began very shortly thereafter at the RAND Corporation in Santa Monica, California. Dantzig’s initial LP work occurred at the Air Force following

Linear programming problems with coefficients of the objective function, depending non linear from one parameter, may be solved using the theory of multiparametric linear programming (cf. Sokolova [3]). The condition under which this theory can be used is a special kind of dependence for each coefficient that consists practically in representing each coefficient as a linear combination with constant and variable factors with a known variation (e.g. whether, variation of different economic factors etc.). One gets subsets of the set on which the curve appearing in such a manner in the parameter space is defined by intersecting this curve with certain maximal polyhedrons, arising by computing the associated multiparametric problem. All elements of such a subset have either the same optimal solution or an unbounded objective function. At least one gives a computing procedure for the described problems.

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.

In constrast to “ordinary” sensitivity analysis in linear programming, the tolerance approach considers simultaneous and independent changes in the objective function coefficients and in the right-hand side terms. This approach yields a maximum tolerance percentage such that, as long as selected coefficients or terms are accurate to within that percentage of their estimated values, the same basis is optimal. In particular, if the objective function coefficients are accurate to within the maximum tolerance percentage of their specified values, then the same solution is optimal.

In this paper, we discusses the problem that involving the conflict fuzzy multi-objective in transportation planning which is one of special case in multi-objective linear programming. To find the simultaneously optimal solution of the problem, we use the Interactive Fuzzy Multi-Objective Linear Programming (IFMOLP) method. This method can reduce the fuzzy multi-objective linear programming problem into deterministic single objective linear programming which can be solved using simplex method. Beside that , with IFMOLP method, decision maker (DM) can establish interactively the goal from the objective function to produce the pareto optimal solution so that.

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.

In this paper, we study general linear programs in which right hand sides are interval numbers. This model is relevant when uncertain and inaccurate factors make difficult the assignment of a single value to each right hand side. When objective function coefficients are interval numbers in a linear program, classical criteria coming from decision theory (like the worst case criterion) are usually applied to determine robust solutions. When the set of feasible solutions is uncertain, we identify a class of linear programs for which these classical approaches are no longer relevant. However, it is possible to compute the worst optimum solution. We study the complexity of this optimization problem when each right hand side is an interval number. Then, we exhibit some duality relationships between the worst optimum solution problem and the best optimum solution to the dual problem.

In the previous chapter, the basics of the simplex and revised simplex algorithms were developed. Degeneracy and convergence issues were addressed. In this chapter we continue with several enhancements to the basic algorithm and related issues. In Section 6.2 we continue the development of sensitivity analysis which was first discussed in Chapters 2 and 3. We show that an analysis of the simplex tableau provides dual variable values, reduced costs and range analysis for the right hand side and objective function, but the simplex algorithm does not provide results which are as reliable as those obtained using projection and inverse projection. In Section 6.3 we develop the dual simplex algorithm. This is the simplex algorithm applied to the dual problem but using the primal simplex tableau. In our earlier development of the simplex algorithm we assumed that all variables had a lower bound of zero and an upper bound of oo. In Section 6.4 we show how to incorporate variables with arbitrary lower and upper bounds into the simplex algorithm. The simplex algorithm requires an initial feasible basis. In Section 6.5 we show how to find an initial basic feasible solution. In Step 3 of the simplex algorithm it is necessary to select a variable to enter the basis. In Section 6.6 we describe several methods for selecting the entering variable. In Section 6.7 we discuss other numerical issues such as tolerances, the minimum ratio test and scaling. Concluding remarks are given in Section 6.8. Exercises are provided in Section 6.9. One topic not covered in this chapter is preprocessing. By preprocessing we mean analyzing the problem prior to solution in order to discover obvious redundancies, tighter variable bounds, eliminate variables and constraints, etc. This is very important in reducing the solution time for solving linear programs. Our discussion of preprocessing linear programs is postponed until Chapter 15.


 The simplex method is a method used to solve linear programming problems with repeated calculations (iterations) that are performed many times until an optimal solution is reached. In solving linear programming problems, it is necessary to determine the constraints and inequalities contained in linear programming. The simplex method can solve linear programming problems involving many inequalities and many variables. Therefore, the purpose of this research is to optimize the production cost of sandals in the Diona Shoes Home Industry. The steps taken to optimize the production cost of sandals include: observing and collecting data, then making a mathematical model, then optimizing production results using the simplex method (iteration), and finally optimizing production results with the Lindo 16.0 software tool.
 Keywords: Diona Shoes, Linear Programming, Simplex Method, Lindo 16.0.

The model of the multi-objective optimal allocation of water resources with government system of water rights is a bi-level programming model. The function of water trade price is difficult to determine, and some factors are interactive such as water price, coefficient of water-saving and objective function, whose relationships are rather complex, so that it is attractive to study the model solution. In this paper, the solution to the bi-level programming model of the optimal allocation of water resources is put forward, in which the upper-level programming is solved by genetic algorithm and the lower-level programming is solved by simplex method, the values about water trade price and coefficient of water-saving are assigned via random-points in a certain range. The results show that the solution is reasonable, and it has been made great progress in ideas and methods to solve this model.

Chapter two - Linear programming

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.

Multivariate probability distributions with given marginals are considered, along with linear functionals, to be minimized or maximized, acting on them. The functionals are supposed to satisfy the Monge or inverse Monge or some higher order convexity property, and they may be only partially known. Existing results in connection with Monge arrays are reformulated and extended in terms of linear programming dual feasible bases. Lower and upper bounds are given for the optimum value as well as for unknown coefficients of the objective function, based on the knowledge of some dual feasible bases and corresponding objective function coefficients. In the two- and three-dimensional cases dual feasible bases are obtained for the problem, where not only the univariate marginals but also the covariances of the pairs of random variables are known.