Early Integer Programming

Problems that reduce to consistency or inconsistency of systems of linear equations or inequalities arise in many divisions of theoretical informatics. The examples are problems in linear programming, machine learning, multicriteria optimization, etc. There exist different stability measures for the property of consistency or inconsistency, and different information constituents are possible (all the input parameters, the coefficient matrix, the vector of constraints). In this paper, variations of all parameters are examined in combination with an additional constraint important in applications, namely, the nonnegativity of feasible points.

Innovative Operational Research practitioner, intellectual leader in the development of systems ideas, and founder of the field of management cybernetics. Born: September 25, 1926 in London, England. Died: August 23, 2002 in Toronto, Canada. Education: D.Sc., University of Sunderland, 2000. Key positions: Production Controller and Head of OR Group, Samuel Fox (1949–1956); Head, Operational Research and Cybernetics, United Steel (1956–1961); Managing Director, SIGMA (1961–1966); Development Director, International Publishing Corporation (1966– 1970); Consultant (1970–2002). Awards and Recognition: Silver Medal, Royal Swedish Academy for Engineering Sciences (1958); Lanchester Prize, Operations Research Society of America (1966); McCullough Plaque of the American Society for Cybernetics (1970); Norbert Weiner Gold Medal, World Organization of Systems and Cybernetics (1984); Lifetime Achievement Award, UK Systems Society (1999); Honorary degrees: Doctor of Laws, Concordia University, Montreal, Canada (1988); Doctor of Economic Sciences, University of St. Gallen, Switzerland (2000). Key OR roles: President, Operational Research Society (1970–1971); President, Society for General Systems Research (1971–1972); President, World Organization of Systems and Cybernetics (1970–the time of his death). Stafford Beer was one of the most remarkable figures that Operational Research (OR) in any country has produced. A charismatic, even flamboyant, character, he founded two major and pioneering OR groups; wrote some of the best books about the subject; and was a world leader in the development of systems ideas. He is widely acknowledged as the founder of the field of International Transactions in Operational Resarch.

The row pivoting method for linear programming

On the problems of linear programming with integral coefficients

We formulate and discuss a model of two-period stochastic optimization which allows for the possibility of the probability distribution over a non-empty finite set of states of nature-exactly one member of which will be realized in the future period- to depend on the control variable chosen in the initial period. In the second and terminal period, the decision maker chooses the value of a control variable vector which along with the control variable vector chosen in the initial period, solve a linear programming problem. The parameters of the linear programming are determined by an uncertain state of nature that is realized at the beginning of the terminal period. In order to obtain specific results, we assume that in the initial period, the decision maker can choose any one alternative from a finite set of alternatives, each such alternative being “a system of linear inequality constraints.” It is further assumed that solution sets of any two different systems are mutually disjoint and the probability distribution of the uncertain components of the state variables that is realized in the second period is determined by the system of linear inequalities chosen in the initial period. We refer to the overall problem as a “two-period stochastic linear programming game”. Our first result for a two-period stochastic linear programming game is that the optimal value for such a game as well as the optimal solutions for each of the finite number of stochastic linear programs can be found by solving a linear programming problem and our second result in this context says that the (optimal) solution for the dual of this linear programming contains a component that determines “at least one” stochastic linear programming problem any solution of which when substituted in its objective function, yields the optimal value of the two-period stochastic linear programming game. There is no need to solve any other subproblem or the primal problem to derive the optimal value of the game. AMS Classifications: 90C05, 90C15, 90C39, 90C46, 91A65 JEL Classifications: C61, C79, D81

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.

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, 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.

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems. Kheirfam and Hasani (Sensitivity analysis for fuzzy linear programming problems with fuzzy variables, Advanced Modeling and Optimization, 12 (2010) 257-272), proposed a method for the sensitivity analysis for fuzzy linear programming (FLP) problems with fuzzy variables. There are some important cases that are not considered by Kheirfam and Hasani. In this paper, those cases are considered and numerical examples are solved.

A suggested algorithm to solve triangular fuzzy rough integer linear programming (TFRILP) problems with α-level is introduced in this paper in order to find rough value optimal solutions and decision rough integer variables, where all parameters and decision variables in the constraints and the objective function are triangular fuzzy rough numbers. In real-life situations, the parameters of a linear programming problem model may not be defined precisely, because of the current market globalization and some other uncontrollable factors. In order to solve this problem, a proper methodology is adopted to solve the TFRILP problems by the slice-sum method with the branch-and-bound technique, through which two fuzzy integers linear programming (FILP) problems with triangular fuzzy interval coefficients and variables were constructed. One of these problems is an FILP problem, where all of its coefficients are the upper approximation interval and represent rather satisfactory solutions; the other is an FILP problem, where all of its coefficients are the lower approximation interval and represent completely satisfactory solutions. Moreover, α-level at α = 0.5 is adopted to find some other rough value optimal solutions and decision rough integer variables. Integer programming is used, since a lot of the linear programming problems require that the decision variables be integers. In addition, the motivation behind this study is to enable the decision makers to make the right decision considering the proposed solutions, while dealing with the uncertain and imprecise data. A flowchart is also provided to illustrate the problem-solving steps. Finally, two numerical examples are given to clarify the obtained results.

Fully intuitionistic fuzzy multi-level linear fractional programming problem

A general concept for solving linear multicriteria programming problems with crisp, fuzzy or stochastic values

ABSTRACTIn this paper, we consider a Fuzzy Stochastic Linear Fractional Programming problem (FSLFP). In this problem, the coefficients and scalars in the objective function are the triangular fuzzy number and technological coefficients and the quantities on the right side of the constraints are fuzzy random variables with the specific distribution. Here we change an FSLFP problem to an equivalent deterministic Multi-objective Linear Fractional Programming (MOLFP) problem. Then by using Fuzzy Mathematical programming approach transformed MOLFP problem is reduced single objective Linear programming (LP) problem. A numerical example is presented to demonstrate the effectiveness of the proposed method.