An Analysis to Treat the Degeneracy of a Basic Feasible Solution in Interval Linear Programming

In this paper an algorithm is given for solving linear fractional functionals programming problems. This method permits to replace in each iteration several basic variables by an equal number of non‐basic ones so as to obtain a new basic feasible solution with the new value of the objective function which is, at least, as good as the old one. To initiate the algorithm, we start with a basic feasible solution. The method of determining a new basic feasible solution, in each iteration, requires the optimum solution of some auxiliary linear fractional functionals program.

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 .

Chapter 18 - Bounded variable problems

Tangent, normal, and visibility cones on Bézier surfaces

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.

In practical power systems, the fault current sensed by the relay depends on location of the fault and network topological changes. Fault current sensed by the relay for near end fault is greater than that for a far end fault. In meshed distribution systems with Distribution Generation (DG), the fault currents are fed from both ends of the line and consequently the currents sensed by the relays at both ends are different. These variable fault currents may lead to miscoordination of directional overcurrent relays (DOCRs). To overcome this problem, the coordination constraints corresponding to the fault currents at different fault locations should be satisfied. This paper proposes a new approach based on Interval Linear Programming (ILP) which is hybridized with Modified Particle Swarm Optimization (MPSO). The basic idea is to convert the inequality constraints of a standard linear programming (LP) problem to interval constraints using theory of ILP and represent the ILP problem as a standard LP problem. This procedure reduces considerable number of coordination constraints and better sub-optimal solution is obtained compared to conventional optimization. The proposed method is tested on IEEE 14 bus test system in microgrid perspective.

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.

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x1,…,xn are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y1,…,yn defined by y1=1−x1,…, yn=1−xn respectively. For a unified notation the variables x1,…,xn and y1,…, yn together will be denoted by z1,…,z2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables zi and zj both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes zi and zj will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments zi =1 and zj =1 which explains the name conflict graph. In other words the non-directed edge between the nodes zi and zj codes the fact that the inequality zi+zj≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

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

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.

Depending on the nature, objectives, and constraints of the decision variables; linear programming, nonlinear programming, integer programming, mixed integer programming etc. can be classified. Extensive research has been conducted to solve all types of these problems in a parametric context. In this paper, to solve optimization problems having uncertainties represented by a single parameter on the objective function, a systematic linearization approach is developed considering the parametric expression as nonlinear. In the proposed approach, the objective function is considered as nonlinear which is converted into linear by using first order Taylor series expansion at the points making the parametric costs zero. Thus, the optimal solution is obtained from the constructed linear programming problem. In this way, by determining the intervals in which the optimal solution changes, the solution of the parametric linear programming problem is obtained. A numerical experiment is illustrated to present the effectiveness of the proposed approach.

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.

Multi-objective linear bi-level multi-follower programming problem (MOBLMFPP) is a special case of two level hierarchical programming problems in which the second level hierarchy includes multiple followers. This paper presents an alternate method based on fuzzy goal programming approach for the solution of multi-objective linear bi-level multi-follower programming (MOLBMFP) problem in which there is no sharing of information among followers. In the proposed FGP model formulation, each of objective functions of each level (leader and follower's level of MOLBMFPP) as well as decision variables at each level are characterised into fuzzy goals. Suitable linear membership functions are defined for each objective function and decision variables. Then minimising the sum of the negative deviational variables of both levels, the highest membership value of each of fuzzy goals is obtained. Existence of compromise optimal solution for MOLBMFPP is established in context of proposed approach. A numerical example is illustrated in support of proposed methodology. A comparative analysis is also carried out in order to show efficiency of proposed method over earlier method for the solution of MOLBMFPP.

This paper presents new linear programming techniques for the design of optimal 1D IIR digital filters. the first method is based on a linear minimax criterion and leads to a linear programming approximation problem whose optimal solution is attained by a new fast algorithm. Next, the linear minimax approximation problem is extended and formulated as two linear integer programming problems which permit the design of 1D IIR digital filters with coefficients of finite word length. the first of these methods is formulated as a 0‐1 integer linear programming problem and its optimal solution is attained rapidly by a new algorithm. This method appears to be suitable for the design of high‐order digital filters. the second method is more general and is formulated as a mixed integer linear programming problem. to solve the design problem efficiently, a new integer tree search algorithm is introduced. the feasibility of the proposed algorithms is illustrated with detailed solutions or numerical examples.