Using complementary slackness property to solve linear programming with fuzzy parameters

The most popular approach to handle the challenge of solving fuzzy linear programming problems is to convert the fuzzy linear programming into the corresponding deterministic linear programming. Mahdavi-Amiri and Nasseri [15,16] developed the fuzzy dual simplex algorithm to fuzzy linear programming with fuzzy parameters. In this paper, we use the complementary slackness to solve it without the need of a simplex tableau.     

Mehar’s method for solving fully fuzzy linear programming problems with L-R fuzzy parameters

To the best of our knowledge, there is no method in literature for solving such fully fuzzy linear programming (FLP) problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers. Also, to propose such a method, there is need to find the product of unrestricted L-R flat fuzzy numbers. However, there is no method in the literature to find the product of unrestricted L-R flat fuzzy numbers.In this paper, firstly the product of unrestricted L-R flat fuzzy numbers is proposed and then with the help of proposed product, a new method (named as Mehar’s method) is proposed for solving fully FLP problems. It is also shown that the fully FLP problems which can be solved by the existing methods can also be solved by the Mehar’s method. However, such fully FLP problems in which some or all the parameters are represented by unrestricted L-R flat fuzzy numbers can be solved by Mehar’s method but can not be solved by any of the existing methods.     

On fuzzy linear programming

Fuzzy multi-objective and fuzzy Goal Programming are discussed in connection with several membership functions which are used to transform the original problem into three equivalent linear programming problems. Existence and uniqueness theorems are given. Fuzzy duality is presented, and an extension of the initial fuzzy problem arises immediately from it.     

Inexact linear programming with generalized resource sets

Inexact linear programs are considered in which the form of the resource set is generalized. Two cases are considered. In the first case closed form solution methods are provided when the resource set as polyhedral. For the general convex resource set no finite representation is apparently possible, although the rudiments of an iterative, approximation algorithm are given along with some illustrations in a numerical example.     

Possibility linear programming with trapezoidal fuzzy numbers

Fuzzy linear programming with trapezoidal fuzzy numbers (TrFNs) is considered and a new method is developed to solve it. In this method, TrFNs are used to capture imprecise or uncertain information for the imprecise objective coefficients and/or the imprecise technological coefficients and/or available resources. The auxiliary multi-objective programming is constructed to solve the corresponding possibility linear programming with TrFNs. The auxiliary multi-objective programming involves four objectives: minimizing the left spread, maximizing the right spread, maximizing the left endpoint of the mode and maximizing the middle point of the mode. Three approaches are proposed to solve the constructed auxiliary multi-objective programming, including optimistic approach, pessimistic approach and linear sum approach based on membership function. An investment example and a transportation problem are presented to demonstrate the implementation process of this method. The comparison analysis shows that the fuzzy linear programming with TrFNs developed in this paper generalizes the possibility linear programming with triangular fuzzy numbers.     

Duality in generalized linear fractional programming

We consider a generalization of a linear fractional program where the maximum of finitely many linear ratios is to be minimized subject to linear constraints. For this Min-Max problem, a dual in the form of a Max-Min problem is introduced and duality relations are established.     

Sensitivity analysis in fuzzy linear programming

In this paper we propose a certain interpretation of a partially fuzzy LP-Problem. This LP-is dualized and the corresponding variables of the Dual are analyzed and an economic interpretation is given.     

Acceptable optimality in linear fractional programming with fuzzy coefficients

Based on the specified grades of satisfaction, we propose two new concepts of (α, β)-acceptable optimal solution and (α, β)-acceptable optimal value of a fuzzy linear fractional programming problem with fuzzy coefficients, and develop a method to compute them. An example is provided to demonstrate the method.     

Finite algorithm for generalized linear multiplicative programming

The nonconvex problem of minimizing the sum of a linear function and the product of two linear functions over a convex polyhedron is considered. A finite algorithm is proposed which either finds a global optimum or shows that the objective function is unbounded from below in the feasible region. This is done by means of a sequence of primal and/or dual simplex iterations.The first author gratefully acknowledges the research support received as Visiting Professor of the Dipartimento di Statistica e Matematica Applicata all' Economia, Universitá di Pisa, Pisa, Italy, Spring 1992.     

Global minimization of a generalized linear multiplicative programming

This article presents a simplicial branch and bound algorithm for globally solving generalized linear multiplicative programming problem (GLMP). Since this problem does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving such problem. In this algorithm, a well known simplicial subdivision is used in the branching procedure and the bound estimation is performed by solving certain linear programs. Convergence of this algorithm is established, and some experiments are reported to show the feasibility of the proposed algorithm.     

Stochastic programming with fuzzy linear partial information on probability distribution

This paper deals with stochastic programming problems where the probability distribution is not explicitly known. We suppose that the probability distribution is defined by crisp or fuzzy inequalities on the probability of the different states of nature. We formulate the problem and present a solution strategy that uses the α-cut technique in order to transform our problem into a stochastic program with linear partial information on probability distribution (SPI). The obtained SPI problem is than solved using two approaches, namely, a chance constrained approach and a recourse approach. For the recourse approach, a modified L-shaped algorithm is designed and illustrated by an example.     

Convex programming with set-inclusive constraints and its applications to generalized linear and fractional programming

Duality results are established in convex programming with the set-inclusive constraints studied by Soyster. The recently developed duality theory for generalized linear programs by Thuente is further generalized and also brought into the framework of Soyster's theory. Convex programming with set-inclusive constraints is further extended to fractional programming.     

A class of multiobjective linear programming model with fuzzy random coefficients

The aim of this paper is to deal with a multiobjective linear programming problem with fuzzy random coefficients. Some crisp equivalent models are presented and a traditional algorithm based on an interactive fuzzy satisfying method is proposed to obtain the decision maker’s satisfying solution. In addition, the technique of fuzzy random simulation is adopted to handle general fuzzy random objective functions and fuzzy random constraints which are usually hard to be converted into their crisp equivalents. Furthermore, combined with the techniques of fuzzy random simulation, a genetic algorithm using the compromise approach is designed for solving a fuzzy random multiobjective programming problem. Finally, illustrative examples are given in order to show the application of the proposed models and algorithms.     

Linear programming with fuzzy parameters: An interactive method resolution

This paper proposes a method for solving linear programming problems where all the coefficients are, in general, fuzzy numbers. We use a fuzzy ranking method to rank the fuzzy objective values and to deal with the inequality relation on constraints. It allows us to work with the concept of feasibility degree. The bigger the feasibility degree is, the worst the objective value will be. We offer the decision-maker (DM) the optimal solution for several different degrees of feasibility. With this information the DM is able to establish a fuzzy goal. We build a fuzzy subset in the decision space whose membership function represents the balance between feasibility degree of constraints and satisfaction degree of the goal. A reasonable solution is the one that has the biggest membership degree to this fuzzy subset. Finally, to illustrate our method, we solve a numerical example.     

Solving multi-level multi-objective linear programming problems through fuzzy goal programming approach

In this paper, two new algorithms are presented to solve multi-level multi-objective linear programming (ML-MOLP) problems through the fuzzy goal programming (FGP) approach. The membership functions for the defined fuzzy goals of all objective functions at all levels are developed in the model formulation of the problem; so also are the membership functions for vectors of fuzzy goals of the decision variables, controlled by decision makers at the top levels. Then the fuzzy goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables and thereby obtain the most satisfactory solution for all decision makers.     

A generalized linear programming model for nurse scheduling

This paper presents a 0–1 column generation model with a resource constrained shortest path auxiliary problem for nurse scheduling. The master problem finds a configuration of individual schedules to satisfy the demand coverage constraints while minimizing salary costs and maximizing both employee preferences and team balance. A feasible solution of the auxiliary problem is an acceptable schedule for a given nurse, with respect to collective agreement requirements such as seniority, workload, rotations and days off. We define a new resource structure in the auxiliary problem in order to take into account the complex collective agreement rules specific to the nurse scheduling problem. This model generalizes further the previous formulations discussed in the literature and can be viewed as a general scheme for complex personnel scheduling problems, especially in the context of organizations which operate around the clock. Solution methods and preliminary test results are discussed.