An Algorithm for Linear Fractional Functionals Programming Problems

It is well known that the G. B. Dantzig’s simplex method is not a good algorithm due to the reason that it is not a polynomial algorithm for use to handle computational complexity; however, it is the popular basic, most effective and widely used way to solve linear programming problems so far. Herewith, various improvements on the simplex procedure have filled many volumes of works. There are two key works that deserve our attention: how to get an initial feasible basis in the form of a unit matrix; how to get the optimal solution as quickly as possible. The basic thought of the simplex method is aimed at the standard form of a linear programming problem:max { }, , , ,C X A X b X b m nT m n = ≥ ≤ (1)Improving the optimal value of the object function via a transform from a basic feasible solution to another basic feasible solution start off at the point where a basic feasible solution is given in its feasible region; and then one obtains the optimal solution or arrives at a judgment that no optimal solution exists. This thought is realized by using a basis transformation in reality. The basis transformation is an operation of iteration by choosing a pivot via determining an entrance variable and an extraction variable according to a given rule. In practical uses, we find that theAlso, assume that B is a basis, and the iteration coefficients A and b are A j( )ai , b = (bi ). By letting XN = 0 , we obtain the corresponding basic solution (Editorial Group of Textbook of Operations Research. 1990, J. L. Xue. 1992, G. H. Wei, J. L. Fu, etc. 1987, Y. L. Niu, 1994)X b b b B b= ⎡⎣ ⎤⎦ = ⎡ ⎣⎢⎤ ⎦⎥1 21 0 00 , , , , , ,…the value of the object function Z C B bB T 1 , andthe criterion σ σ σ[ ,σ , , ]1 2 n T .

When coefficients in the objective function cannot be precisely determined, the optimal solution is fluctuated by the realisation of coefficients. Therefore, analysing the stability of an optimal solution becomes essential. Although the robustness analysis of an optimal basic solution has been developed successfully so far, it becomes complex when the solution contains degeneracy. This study is devoted to overcoming the difficulty caused by the degeneracy in a linear programming problem with interval objective coefficients. We focus on the tangent cone of a degenerate basic feasible solution since the belongingness of the objective coefficient vector to its associated normal cone assures the solution’s optimality. We decompose the normal cone by its associated tangent cone to a direct union of subspaces. Several propositions related to the proposed approach are given. To demonstrate the significance of the decomposition, we consider the case where the dimension of the subspace is one. We examine the obtained propositions by numerical examples with comparisons to the conventional techniques.KeywordsInterval linear programmingDegeneracyPolyhedral convex coneTangent coneBasic space

Chapter 18 - Bounded variable problems

There are two important approaches based on linear ranking functions for solving linear programming problems with cost coefficients as an auxiliary problem to obtain a fuzzy solution of fuzzy variable linear programming problem. The first approach uses the primal simplex method that assumes an initial primal feasible basic solution is at hand. The second approach is based on dual simplex method that begins with a basic dual feasible basic solution and proceeds by pivoting through a series of dual basic solutions until the associated complementary primal basic fuzzy solution is feasible. In this paper, we propose a new method called the primal-dual algorithm, which is similar to the dual simplex method and begins with dual feasibility and proceeds to obtain primal feasibility while maintaining complementary slackness. An important difference between the dual simplex method and the primal-dual method is that the primal-dual algorithm does not require a dual feasible solution to be basic. This algorithm is useful specially for solving minimum fuzzy cost flow problem in which finding an initial dual feasible solution turns out to be a trivial task.

Chapter 16 - Variants of the simplex method

The idea of the simplex method is to proceed from one basic feasible solution (that is, one extreme point) of the constraint set of a problem in standard form to another, in such a way as to continually decrease the value of the objective function until a minimum is reached. The results of Chap. 2 assure us that it is sufficient to consider only basic feasible solutions in our search for an optimal feasible solution. This chapter demonstrates that an efficient method for moving among basic solutions to the minimum can be constructed.

Improvement and its computer implementation of an artificial-free simplex-type algorithm by Arsham

It is shown that solutions of linear inequalities, linear programs and certain linear complementarity problems (e.g. those with P-matrices or Z-matrices but not semidefinite matrices) are Lipschitz continuous with respect to changes in the right-hand side data of the problem. Solutions of linear programs are not Lipschitz continuous with respect to the coefficients of the objective function. The Lipschitz constant given here is a generalization of the role played by the norm of the inverse of a nonsingular matrix in bounding the perturbation of the solution of a system of equations in terms of a right-hand side perturbation.

This paper introduces a new approach to optimize the cost per unit of product for the Transportation Problem to achieve better outcomes. We present Basic Feasible Solution (BFS) approach compromised of five main steps: (1) Create a Matrix A = mod |Supply(si)-Demand(dj)| (2) Add the cost of each cell of cost matrix C with corresponding elements of Matrix A and Create Matrix B. (3) Mark number in Ascending order of each elements of Matrix B from 1 to mxn (4) If si ≠ dj, then ζ = |small(si, dj)|, else ζ = si or dj, Assign ζ in Matrix B to smallest number from 1 to mxn, and cut the rest of elements of row ζ or column ζ and subtract ζ from other than selected; and (5) Repeat step 4, until all the supply and demand become zero. This solution approach finds the basic feasible solution of TP with the same complexity for solving Vogel’s approximation method (VAM).KeywordsMount Order Method (MOM)Transportation Problem (TP)Basic Feasible Solution (BFS)Optimal Solution (OS)Stepping stone method

The relationship between solutions of the full master program of the Dantzig-Wolfe decomposition routine and the originally given problem is considered, and a criterion is established which states whether a basic feasible solution of the former corresponds to one of the latter.

In this research, a new and efficient approach of finding an initial basic feasible solution to transportation problems is proposed. The proposed approach is named “Inverse Coefficient of Variation Method (ICVM)”, and the method is illustrated with seven numerical examples. Six existing methods; North West Corner Method (NWCM), Column Minimum Method (CMM), Least Cost Method (LCM), Row Minimum Method (RMM),Vogel’s Approximation Method (VAM), and Allocation Table Method (ATM) were compared with the proposed approach. It can be said conclusively that the proposed Inverse Coefficient of Variation Method (ICVM) provides an improved Initial Basic Feasible Solution to all the transportation problems used in the experiment. Further, the new method leads to the optimal solution to many of the problems considered.

An approximate method for structural optimisation

The aim of this paper is twofold. Firstly, to define a solution concept of Pareto-optimality for a multi-objective flexible linear programming (MOFLP) problem (or multi-objective fuzzy linear programming problem) and design a method to extract a Pareto-optimal solution of MOFLP problem from the set of optimal solutions of equivalent optimization problem formulated by Dubey and Mehra (2013). Secondly, to extend this study to multi-objective linear programming problem involving hard and flexible constraints with interval uncertainty. A flexible constraint with interval uncertainty generalizes a flexible constraint by allowing preferences to be expressed in the form of intervals. An optimistic-pessimistic approach is proposed to solve multi-objective flexible linear programming with interval uncertainty (MOFLPIU) using an interval-valued fuzzy set representation and the Hurwicz optimism-pessimism criterion.

Computational behavior of a feasible direction method for linear programming

In this paper we develop a new algorithm for the initial fuzzy basic feasible solution to a solid transportation problem with imprecise parameters. All the decision variables are assumed to be triangular fuzzy numbers. Using parametric form of triangular fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using proposed algorithm we obtained the initial basic fuzzy feasible solution to the problem without changing to its crisp equivalent form. We further discuss the optimality by applying the modified distribution method. A numerical example is solved to show the effectiveness of the proposed algorithm.